Creation mechanism of Devil’s Staircase surface and unstable and stable periodic orbits in the anisotropic Kepler problem

Abstract

In the two-dimensional Anisotropic Kepler Problem (AKP), a longstanding question concerns the uniqueness of an unstable periodic orbit (PO) for a given binary code (modulo symmetry equivalence). In this paper, a finite-level (N) surface defined by the binary coding of the orbit is considered over the initial-value domain D0. A tiling of D0 by base ribbons of the surface steps is shown to be proper, i.e., the surface height increases monotonously when the ribbons are traversed from left to right. The mechanism of creating a level-(N+1) tiling from the level-N tiling is clarified in the course of the proof. There are two possible cases depending on the code and the anisotropy. (A) Every ribbon shrinks to a line as . Here, the uniqueness holds. (B) When future (F) and past (P) ribbons become tangential to each other, they escape from the shrinking. Then, the initial values of a stable PO (S) and an unstable PO (U) sharing the same code co-exist inside the overlap of the F and P non-shrinking ribbons. This corresponds to Broucke’s PO. When the anisotropy is high, only case A is observed; however, as the anisotropy decreases, a bifurcation of the form occurs along with the emergence of a non-shrinking ribbon. (Here, R and NR denote self-retracing and non-self-retracing POs, respectively). We conjecture that, from a classification based on topology and symmetry, case B occurs only for odd-rank, Y-symmetric POs. We report two applications. First, the classification is applied successfully to the successive bifurcation of a high-rank PO (n = 15), where the above bifurcation is followed by . Second, enhancing the sensitivity to the co-existence of S and U POs through ribbon tiling, we examine the high-anisotropy region. A new symmetry-type POs (O-type) are found and, at γ = 0.2, all POs are shown to be unstable and unique. An investigation of 13648 POs at rank 10 verifies that Gutzwiller’s action formula works with amazing accuracy.